To solve the problem, we need to analyze the given values of \( \sin \theta_1 \), \( \cos^2 \theta_2 \), and \( \tan \theta_3 \), which are \( \frac{1}{2} \), \( -\frac{1}{2} \), and \( 3 \) (not in order). We are to choose the incorrect statement about these values.
### Step-by-Step Solution:
1. **Identify the Possible Values**:
- We have three values: \( \frac{1}{2} \), \( -\frac{1}{2} \), and \( 3 \).
- These values correspond to \( \sin \theta_1 \), \( \cos^2 \theta_2 \), and \( \tan \theta_3 \).
2. **Determine Valid Ranges**:
- The sine function, \( \sin \theta \), can take values from \( -1 \) to \( 1 \). Therefore, \( \sin \theta_1 \) can be \( \frac{1}{2} \) or \( -\frac{1}{2} \).
- The cosine squared function, \( \cos^2 \theta \), must be non-negative and can take values from \( 0 \) to \( 1 \). Thus, \( \cos^2 \theta_2 \) cannot be \( -\frac{1}{2} \) or \( 3 \).
- The tangent function, \( \tan \theta \), can take any real number value, so \( \tan \theta_3 \) can be \( 3 \).
3. **Assign Values**:
- Since \( \cos^2 \theta_2 \) cannot be negative, we assign:
- \( \cos^2 \theta_2 = \frac{1}{2} \)
- This means \( \sin \theta_1 = -\frac{1}{2} \) (since it can be either \( \frac{1}{2} \) or \( -\frac{1}{2} \)).
- Consequently, \( \tan \theta_3 = 3 \).
4. **Evaluate Statements**:
- Now we evaluate the statements based on the assigned values:
- **Statement A**: \( \tan \theta_3 \) could be \( -\frac{1}{2} \) (Incorrect, since \( \tan \theta_3 = 3 \)).
- **Statement B**: \( \sin \theta_1 \) cannot be \( 3 \) (Correct, since \( \sin \theta_1 = -\frac{1}{2} \)).
- **Statement C**: \( \cos^2 \theta_2 \) cannot be \( -\frac{1}{2} \) (Correct, since \( \cos^2 \theta_2 = \frac{1}{2} \)).
- **Statement D**: \( \cos^2 \theta_2 \) could be \( 3 \) (Incorrect, since \( \cos^2 \theta_2 = \frac{1}{2} \)).
5. **Conclusion**:
- The incorrect statements are A and D. However, since we need to choose one incorrect statement, we can select either A or D based on the context of the question.
### Final Answer:
The incorrect statement is **A**: \( \tan \theta_3 \) could be \( -\frac{1}{2} \).
